The order of zeros of automorphic L -functions and the value of L-functions

• Project/Area Number: 23740020
• Research Category: Grant-in-Aid for Young Scientists (B)
• Allocation Type: Multi-year Fund
• Research Field: Algebra
• Research Institution: Nara Women's University
• Principal Investigator: Umegaki Yumiko 奈良女子大学, 自然科学系, 准教授 (80372689)
• Project Period (FY): 2011-04-28 – 2019-03-31
• Project Status: Completed (Fiscal Year 2018)
• Budget Amount: ¥4,030,000 (Direct Cost: ¥3,100,000、Indirect Cost: ¥930,000); Fiscal Year 2014: ¥910,000 (Direct Cost: ¥700,000、Indirect Cost: ¥210,000); Fiscal Year 2013: ¥910,000 (Direct Cost: ¥700,000、Indirect Cost: ¥210,000); Fiscal Year 2012: ¥910,000 (Direct Cost: ¥700,000、Indirect Cost: ¥210,000); Fiscal Year 2011: ¥1,300,000 (Direct Cost: ¥1,000,000、Indirect Cost: ¥300,000)
• Keywords: 特殊値 / 類数 / 値分布 / analytic rank / class number / 整数論 / 数論

Outline of Final Research Achievements

The present study evolved the methods for the study of the value distribution of Riemann zeta function, Dirichlet L-function and the logarithm of them, and applied this method to the value distribution of the logarithm of the automorphic L functions.
For the composite fields with large class numbers, the present study established the method to prove that there exist infinite many such fields. And it was proved for all types of quatic number fields. Moreover the possibility of it was proved for bycubic number fields.

Academic Significance and Societal Importance of the Research Achievements

この研究は整数論の諸問題に対して様々な応用が考えられる。保型 L 関数の場合、関数等式の折り返しの点における零点の位数は解析的階数として楕円曲線の有理点が成す有限生成アーベル群の階数と関連する。この階数を解析的に研究する新しい手法を導入した。また、ディリクレ L 関数の特殊値は代数体の類数と関係するため、古くから研究がされているが、合成体に関して類数の大きなものが無限に存在することを議論する方法論を確立した。これらの結果やその手法は今後の整数論の研究においてとても効果的であると言える。

Research Products

• [Int'l Joint Research] Universite de Franche-Comte(France)
• [Journal Article] Bicubic number fields with large class numbers (2020)
  • Author(s): Yasuko Morita, Atsuki Umegaki and Yumiko Umegaki
  • Journal Title: Various Aspects of Multiple Zeta Functions in honor of Professor Kohji Matsumoto's 60th birthday, Adv. Stud. Pure Math. Volume: 印刷中
  • Peer Reviewed
• [Journal Article] On the density function for the value-distribution of automorphic L-functions (2019)
  • Author(s): Kohji Matsumoto and Yumiko Umegaki
  • Journal Title: Journal of Number Theory Volume: 198 Pages: 176-199
  • DOI: 10.1016/j.jnt.2018.10.008
  • Peer Reviewed
• [Journal Article] On the value-distribution of symmetric power L-functions (2019)
  • Author(s): Kohji Matsumoto and Yumiko Umegaki
  • Journal Title: lecture note series of the Ramanujan Mathematical Society Volume: 印刷中
  • Peer Reviewed
• [Journal Article] On the value-distribution of the difference between logarithms of two symmetric power L-functions (2018)
  • Author(s): Matsumoto Kohji、Umegaki Yumiko
  • Journal Title: International Journal of Number Theory Volume: 14 Issue: 07 Pages: 2045-2081
  • DOI: 10.1142/s1793042118501245
  • Peer Reviewed
• [Journal Article] On the value-distribution of the difference between logarithms of two symmetric power L-functions (2018)
  • Author(s): Kohji Matsumoto, Yumiko Umegaki
  • Journal Title: International Journal of Number Theory Volume: 印刷中
  • Peer Reviewed
• [Presentation] On the value-distribution of the difference between logarithms of two symmetric power L-functions (2017)
  • Author(s): Yumiko Umegaki
  • Organizer: Various Aspects of Multiple Zeta Function
  • Int'l Joint Research
• [Presentation] On the value-distribution of the difference between logarithms of two symmetric power L-functions (2016)
  • Author(s): Kohji Matsumoto and Yumiko Umegaki
  • Organizer: Aspects on Universality 2016
  • Place of Presentation: University of Wurzburg, Germany
  • Year and Date: 2016-05-06
  • Int'l Joint Research